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<div class="title">opennurbs_matrix.h</div>  </div>
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<div class="contents">
<div class="fragment"><div class="line"><a name="l00001"></a><span class="lineno">    1</span>&#160;<span class="comment">/* $NoKeywords: $ */</span></div>
<div class="line"><a name="l00002"></a><span class="lineno">    2</span>&#160;<span class="comment">/*</span></div>
<div class="line"><a name="l00003"></a><span class="lineno">    3</span>&#160;<span class="comment">//</span></div>
<div class="line"><a name="l00004"></a><span class="lineno">    4</span>&#160;<span class="comment">// Copyright (c) 1993-2012 Robert McNeel &amp; Associates. All rights reserved.</span></div>
<div class="line"><a name="l00005"></a><span class="lineno">    5</span>&#160;<span class="comment">// OpenNURBS, Rhinoceros, and Rhino3D are registered trademarks of Robert</span></div>
<div class="line"><a name="l00006"></a><span class="lineno">    6</span>&#160;<span class="comment">// McNeel &amp; Associates.</span></div>
<div class="line"><a name="l00007"></a><span class="lineno">    7</span>&#160;<span class="comment">//</span></div>
<div class="line"><a name="l00008"></a><span class="lineno">    8</span>&#160;<span class="comment">// THIS SOFTWARE IS PROVIDED &quot;AS IS&quot; WITHOUT EXPRESS OR IMPLIED WARRANTY.</span></div>
<div class="line"><a name="l00009"></a><span class="lineno">    9</span>&#160;<span class="comment">// ALL IMPLIED WARRANTIES OF FITNESS FOR ANY PARTICULAR PURPOSE AND OF</span></div>
<div class="line"><a name="l00010"></a><span class="lineno">   10</span>&#160;<span class="comment">// MERCHANTABILITY ARE HEREBY DISCLAIMED.</span></div>
<div class="line"><a name="l00011"></a><span class="lineno">   11</span>&#160;<span class="comment">//              </span></div>
<div class="line"><a name="l00012"></a><span class="lineno">   12</span>&#160;<span class="comment">// For complete openNURBS copyright information see &lt;http://www.opennurbs.org&gt;.</span></div>
<div class="line"><a name="l00013"></a><span class="lineno">   13</span>&#160;<span class="comment">//</span></div>
<div class="line"><a name="l00015"></a><span class="lineno">   15</span>&#160;<span class="comment">*/</span></div>
<div class="line"><a name="l00016"></a><span class="lineno">   16</span>&#160; </div>
<div class="line"><a name="l00017"></a><span class="lineno">   17</span>&#160;<span class="preprocessor">#if !defined(ON_MATRIX_INC_)</span></div>
<div class="line"><a name="l00018"></a><span class="lineno">   18</span>&#160;<span class="preprocessor">#define ON_MATRIX_INC_</span></div>
<div class="line"><a name="l00019"></a><span class="lineno">   19</span>&#160; </div>
<div class="line"><a name="l00020"></a><span class="lineno">   20</span>&#160;<span class="keyword">class </span><a class="code" href="class_o_n___xform.html">ON_Xform</a>;</div>
<div class="line"><a name="l00021"></a><span class="lineno">   21</span>&#160; </div>
<div class="line"><a name="l00022"></a><span class="lineno"><a class="line" href="class_o_n___matrix.html">   22</a></span>&#160;<span class="keyword">class </span>ON_CLASS <a class="code" href="class_o_n___matrix.html">ON_Matrix</a></div>
<div class="line"><a name="l00023"></a><span class="lineno">   23</span>&#160;{</div>
<div class="line"><a name="l00024"></a><span class="lineno">   24</span>&#160;<span class="keyword">public</span>:</div>
<div class="line"><a name="l00025"></a><span class="lineno">   25</span>&#160;  <a class="code" href="class_o_n___matrix.html">ON_Matrix</a>();</div>
<div class="line"><a name="l00026"></a><span class="lineno">   26</span>&#160;  <a class="code" href="class_o_n___matrix.html">ON_Matrix</a>( </div>
<div class="line"><a name="l00027"></a><span class="lineno">   27</span>&#160;    <span class="keywordtype">int</span> row_count,</div>
<div class="line"><a name="l00028"></a><span class="lineno">   28</span>&#160;    <span class="keywordtype">int</span> col_count</div>
<div class="line"><a name="l00029"></a><span class="lineno">   29</span>&#160;     );</div>
<div class="line"><a name="l00030"></a><span class="lineno">   30</span>&#160;  <a class="code" href="class_o_n___matrix.html">ON_Matrix</a>( <span class="comment">// see ON_Matrix::Create(int,int,int,int) for details</span></div>
<div class="line"><a name="l00031"></a><span class="lineno">   31</span>&#160;     <span class="keywordtype">int</span>, <span class="comment">// first valid row index</span></div>
<div class="line"><a name="l00032"></a><span class="lineno">   32</span>&#160;     <span class="keywordtype">int</span>, <span class="comment">// last valid row index</span></div>
<div class="line"><a name="l00033"></a><span class="lineno">   33</span>&#160;     <span class="keywordtype">int</span>, <span class="comment">// first valid column index</span></div>
<div class="line"><a name="l00034"></a><span class="lineno">   34</span>&#160;     <span class="keywordtype">int</span>  <span class="comment">// last valid column index</span></div>
<div class="line"><a name="l00035"></a><span class="lineno">   35</span>&#160;     );</div>
<div class="line"><a name="l00036"></a><span class="lineno">   36</span>&#160;  <a class="code" href="class_o_n___matrix.html">ON_Matrix</a>( <span class="keyword">const</span> <a class="code" href="class_o_n___xform.html">ON_Xform</a>&amp; );</div>
<div class="line"><a name="l00037"></a><span class="lineno">   37</span>&#160;  <a class="code" href="class_o_n___matrix.html">ON_Matrix</a>( <span class="keyword">const</span> <a class="code" href="class_o_n___matrix.html">ON_Matrix</a>&amp; );</div>
<div class="line"><a name="l00038"></a><span class="lineno">   38</span>&#160; </div>
<div class="line"><a name="l00039"></a><span class="lineno">   39</span>&#160;  <span class="comment">/*</span></div>
<div class="line"><a name="l00040"></a><span class="lineno">   40</span>&#160;<span class="comment">  Description:</span></div>
<div class="line"><a name="l00041"></a><span class="lineno">   41</span>&#160;<span class="comment">    This constructor is for experts who have storage for a matrix</span></div>
<div class="line"><a name="l00042"></a><span class="lineno">   42</span>&#160;<span class="comment">    and need to use it in ON_Matrix form.</span></div>
<div class="line"><a name="l00043"></a><span class="lineno">   43</span>&#160;<span class="comment">  Parameters:</span></div>
<div class="line"><a name="l00044"></a><span class="lineno">   44</span>&#160;<span class="comment">    row_count - [in]</span></div>
<div class="line"><a name="l00045"></a><span class="lineno">   45</span>&#160;<span class="comment">    col_count - [in]</span></div>
<div class="line"><a name="l00046"></a><span class="lineno">   46</span>&#160;<span class="comment">    M - [in]</span></div>
<div class="line"><a name="l00047"></a><span class="lineno">   47</span>&#160;<span class="comment">    bDestructorFreeM - [in]</span></div>
<div class="line"><a name="l00048"></a><span class="lineno">   48</span>&#160;<span class="comment">      If true, ~ON_Matrix will call onfree(M).</span></div>
<div class="line"><a name="l00049"></a><span class="lineno">   49</span>&#160;<span class="comment">      If false, caller is managing M&#39;s memory.</span></div>
<div class="line"><a name="l00050"></a><span class="lineno">   50</span>&#160;<span class="comment">  Remarks:</span></div>
<div class="line"><a name="l00051"></a><span class="lineno">   51</span>&#160;<span class="comment">    ON_Matrix functions that increase the value of row_count or col_count</span></div>
<div class="line"><a name="l00052"></a><span class="lineno">   52</span>&#160;<span class="comment">    will fail on a matrix created with this constructor.</span></div>
<div class="line"><a name="l00053"></a><span class="lineno">   53</span>&#160;<span class="comment">  */</span></div>
<div class="line"><a name="l00054"></a><span class="lineno">   54</span>&#160;  <a class="code" href="class_o_n___matrix.html">ON_Matrix</a>(</div>
<div class="line"><a name="l00055"></a><span class="lineno">   55</span>&#160;    <span class="keywordtype">int</span> row_count,</div>
<div class="line"><a name="l00056"></a><span class="lineno">   56</span>&#160;    <span class="keywordtype">int</span> col_count,</div>
<div class="line"><a name="l00057"></a><span class="lineno">   57</span>&#160;    <span class="keywordtype">double</span>** M,</div>
<div class="line"><a name="l00058"></a><span class="lineno">   58</span>&#160;    <span class="keywordtype">bool</span> bDestructorFreeM</div>
<div class="line"><a name="l00059"></a><span class="lineno">   59</span>&#160;    );</div>
<div class="line"><a name="l00060"></a><span class="lineno">   60</span>&#160; </div>
<div class="line"><a name="l00061"></a><span class="lineno">   61</span>&#160;  <span class="keyword">virtual</span> ~<a class="code" href="class_o_n___matrix.html">ON_Matrix</a>();</div>
<div class="line"><a name="l00062"></a><span class="lineno">   62</span>&#160;  <span class="keywordtype">void</span> EmergencyDestroy(); <span class="comment">// call if memory pool used matrix by becomes invalid</span></div>
<div class="line"><a name="l00063"></a><span class="lineno">   63</span>&#160; </div>
<div class="line"><a name="l00064"></a><span class="lineno">   64</span>&#160;  <span class="comment">// ON_Matrix[i][j] = value at row i and column j</span></div>
<div class="line"><a name="l00065"></a><span class="lineno">   65</span>&#160;  <span class="comment">//           0 &lt;= i &lt; RowCount()</span></div>
<div class="line"><a name="l00066"></a><span class="lineno">   66</span>&#160;  <span class="comment">//           0 &lt;= j &lt; ColCount()</span></div>
<div class="line"><a name="l00067"></a><span class="lineno">   67</span>&#160;  <span class="keywordtype">double</span>* operator[](<span class="keywordtype">int</span>);</div>
<div class="line"><a name="l00068"></a><span class="lineno">   68</span>&#160;  <span class="keyword">const</span> <span class="keywordtype">double</span>* operator[](<span class="keywordtype">int</span>) <span class="keyword">const</span>;</div>
<div class="line"><a name="l00069"></a><span class="lineno">   69</span>&#160; </div>
<div class="line"><a name="l00070"></a><span class="lineno">   70</span>&#160;  <a class="code" href="class_o_n___matrix.html">ON_Matrix</a>&amp; operator=(<span class="keyword">const</span> <a class="code" href="class_o_n___matrix.html">ON_Matrix</a>&amp;);</div>
<div class="line"><a name="l00071"></a><span class="lineno">   71</span>&#160;  <a class="code" href="class_o_n___matrix.html">ON_Matrix</a>&amp; operator=(<span class="keyword">const</span> <a class="code" href="class_o_n___xform.html">ON_Xform</a>&amp;);</div>
<div class="line"><a name="l00072"></a><span class="lineno">   72</span>&#160; </div>
<div class="line"><a name="l00073"></a><span class="lineno">   73</span>&#160;  <span class="keywordtype">bool</span> IsValid() <span class="keyword">const</span>;</div>
<div class="line"><a name="l00074"></a><span class="lineno">   74</span>&#160;  <span class="keywordtype">int</span> IsSquare() <span class="keyword">const</span>; <span class="comment">// returns 0 for no and m_row_count (= m_col_count) for yes</span></div>
<div class="line"><a name="l00075"></a><span class="lineno">   75</span>&#160;  <span class="keywordtype">int</span> RowCount() <span class="keyword">const</span>;</div>
<div class="line"><a name="l00076"></a><span class="lineno">   76</span>&#160;  <span class="keywordtype">int</span> ColCount() <span class="keyword">const</span>;</div>
<div class="line"><a name="l00077"></a><span class="lineno">   77</span>&#160;  <span class="keywordtype">int</span> MinCount() <span class="keyword">const</span>; <span class="comment">// smallest of row and column count</span></div>
<div class="line"><a name="l00078"></a><span class="lineno">   78</span>&#160;  <span class="keywordtype">int</span> MaxCount() <span class="keyword">const</span>; <span class="comment">// largest of row and column count</span></div>
<div class="line"><a name="l00079"></a><span class="lineno">   79</span>&#160; </div>
<div class="line"><a name="l00080"></a><span class="lineno">   80</span>&#160;  <span class="keywordtype">void</span> RowScale(<span class="keywordtype">int</span>,<span class="keywordtype">double</span>); </div>
<div class="line"><a name="l00081"></a><span class="lineno">   81</span>&#160;  <span class="keywordtype">void</span> ColScale(<span class="keywordtype">int</span>,<span class="keywordtype">double</span>);</div>
<div class="line"><a name="l00082"></a><span class="lineno">   82</span>&#160;  <span class="keywordtype">void</span> RowOp(<span class="keywordtype">int</span>,<span class="keywordtype">double</span>,<span class="keywordtype">int</span>);</div>
<div class="line"><a name="l00083"></a><span class="lineno">   83</span>&#160;  <span class="keywordtype">void</span> ColOp(<span class="keywordtype">int</span>,<span class="keywordtype">double</span>,<span class="keywordtype">int</span>);</div>
<div class="line"><a name="l00084"></a><span class="lineno">   84</span>&#160; </div>
<div class="line"><a name="l00085"></a><span class="lineno">   85</span>&#160;  <span class="keywordtype">bool</span> Create(</div>
<div class="line"><a name="l00086"></a><span class="lineno">   86</span>&#160;     <span class="keywordtype">int</span>, <span class="comment">// number of rows</span></div>
<div class="line"><a name="l00087"></a><span class="lineno">   87</span>&#160;     <span class="keywordtype">int</span>  <span class="comment">// number of columns</span></div>
<div class="line"><a name="l00088"></a><span class="lineno">   88</span>&#160;     );</div>
<div class="line"><a name="l00089"></a><span class="lineno">   89</span>&#160; </div>
<div class="line"><a name="l00090"></a><span class="lineno">   90</span>&#160;  <span class="keywordtype">bool</span> Create( <span class="comment">// E.g., Create(1,5,1,7) creates a 5x7 sized matrix that with</span></div>
<div class="line"><a name="l00091"></a><span class="lineno">   91</span>&#160;               <span class="comment">// &quot;top&quot; row = m[1][1],...,m[1][7] and &quot;bottom&quot; row</span></div>
<div class="line"><a name="l00092"></a><span class="lineno">   92</span>&#160;               <span class="comment">// = m[5][1],...,m[5][7].  The result of Create(0,m,0,n) is</span></div>
<div class="line"><a name="l00093"></a><span class="lineno">   93</span>&#160;               <span class="comment">// identical to the result of Create(m+1,n+1).</span></div>
<div class="line"><a name="l00094"></a><span class="lineno">   94</span>&#160;     <span class="keywordtype">int</span>, <span class="comment">// first valid row index</span></div>
<div class="line"><a name="l00095"></a><span class="lineno">   95</span>&#160;     <span class="keywordtype">int</span>, <span class="comment">// last valid row index</span></div>
<div class="line"><a name="l00096"></a><span class="lineno">   96</span>&#160;     <span class="keywordtype">int</span>, <span class="comment">// first valid column index</span></div>
<div class="line"><a name="l00097"></a><span class="lineno">   97</span>&#160;     <span class="keywordtype">int</span>  <span class="comment">// last valid column index</span></div>
<div class="line"><a name="l00098"></a><span class="lineno">   98</span>&#160;     );</div>
<div class="line"><a name="l00099"></a><span class="lineno">   99</span>&#160; </div>
<div class="line"><a name="l00100"></a><span class="lineno">  100</span>&#160;  <span class="comment">/*</span></div>
<div class="line"><a name="l00101"></a><span class="lineno">  101</span>&#160;<span class="comment">  Description:</span></div>
<div class="line"><a name="l00102"></a><span class="lineno">  102</span>&#160;<span class="comment">    This constructor is for experts who have storage for a matrix</span></div>
<div class="line"><a name="l00103"></a><span class="lineno">  103</span>&#160;<span class="comment">    and need to use it in ON_Matrix form.</span></div>
<div class="line"><a name="l00104"></a><span class="lineno">  104</span>&#160;<span class="comment">  Parameters:</span></div>
<div class="line"><a name="l00105"></a><span class="lineno">  105</span>&#160;<span class="comment">    row_count - [in]</span></div>
<div class="line"><a name="l00106"></a><span class="lineno">  106</span>&#160;<span class="comment">    col_count - [in]</span></div>
<div class="line"><a name="l00107"></a><span class="lineno">  107</span>&#160;<span class="comment">    M - [in]</span></div>
<div class="line"><a name="l00108"></a><span class="lineno">  108</span>&#160;<span class="comment">    bDestructorFreeM - [in]</span></div>
<div class="line"><a name="l00109"></a><span class="lineno">  109</span>&#160;<span class="comment">      If true, ~ON_Matrix will call onfree(M).</span></div>
<div class="line"><a name="l00110"></a><span class="lineno">  110</span>&#160;<span class="comment">      If false, caller is managing M&#39;s memory.</span></div>
<div class="line"><a name="l00111"></a><span class="lineno">  111</span>&#160;<span class="comment">  Remarks:</span></div>
<div class="line"><a name="l00112"></a><span class="lineno">  112</span>&#160;<span class="comment">    ON_Matrix functions that increase the value of row_count or col_count</span></div>
<div class="line"><a name="l00113"></a><span class="lineno">  113</span>&#160;<span class="comment">    will fail on a matrix created with this constructor.</span></div>
<div class="line"><a name="l00114"></a><span class="lineno">  114</span>&#160;<span class="comment">  */</span></div>
<div class="line"><a name="l00115"></a><span class="lineno">  115</span>&#160;  <span class="keywordtype">bool</span> Create(</div>
<div class="line"><a name="l00116"></a><span class="lineno">  116</span>&#160;    <span class="keywordtype">int</span> row_count,</div>
<div class="line"><a name="l00117"></a><span class="lineno">  117</span>&#160;    <span class="keywordtype">int</span> col_count,</div>
<div class="line"><a name="l00118"></a><span class="lineno">  118</span>&#160;    <span class="keywordtype">double</span>** M,</div>
<div class="line"><a name="l00119"></a><span class="lineno">  119</span>&#160;    <span class="keywordtype">bool</span> bDestructorFreeM</div>
<div class="line"><a name="l00120"></a><span class="lineno">  120</span>&#160;    );</div>
<div class="line"><a name="l00121"></a><span class="lineno">  121</span>&#160; </div>
<div class="line"><a name="l00122"></a><span class="lineno">  122</span>&#160; </div>
<div class="line"><a name="l00123"></a><span class="lineno">  123</span>&#160;  <span class="keywordtype">void</span> Destroy();</div>
<div class="line"><a name="l00124"></a><span class="lineno">  124</span>&#160; </div>
<div class="line"><a name="l00125"></a><span class="lineno">  125</span>&#160;  <span class="keywordtype">void</span> Zero();</div>
<div class="line"><a name="l00126"></a><span class="lineno">  126</span>&#160; </div>
<div class="line"><a name="l00127"></a><span class="lineno">  127</span>&#160;  <span class="keywordtype">void</span> SetDiagonal(<span class="keywordtype">double</span>); <span class="comment">// sets diagonal value and zeros off diagonal values</span></div>
<div class="line"><a name="l00128"></a><span class="lineno">  128</span>&#160;  <span class="keywordtype">void</span> SetDiagonal(<span class="keyword">const</span> <span class="keywordtype">double</span>*); <span class="comment">// sets diagonal values and zeros off diagonal values</span></div>
<div class="line"><a name="l00129"></a><span class="lineno">  129</span>&#160;  <span class="keywordtype">void</span> SetDiagonal(<span class="keywordtype">int</span>, <span class="keyword">const</span> <span class="keywordtype">double</span>*); <span class="comment">// sets size to count x count and diagonal values and zeros off diagonal values</span></div>
<div class="line"><a name="l00130"></a><span class="lineno">  130</span>&#160;  <span class="keywordtype">void</span> SetDiagonal(<span class="keyword">const</span> <a class="code" href="class_o_n___simple_array.html">ON_SimpleArray&lt;double&gt;</a>&amp;); <span class="comment">// sets size to length X lengthdiagonal values and zeros off diagonal values</span></div>
<div class="line"><a name="l00131"></a><span class="lineno">  131</span>&#160; </div>
<div class="line"><a name="l00132"></a><span class="lineno">  132</span>&#160;  <span class="keywordtype">bool</span> Transpose();</div>
<div class="line"><a name="l00133"></a><span class="lineno">  133</span>&#160; </div>
<div class="line"><a name="l00134"></a><span class="lineno">  134</span>&#160;  <span class="keywordtype">bool</span> SwapRows( <span class="keywordtype">int</span>, <span class="keywordtype">int</span> ); <span class="comment">// ints are row indices to swap</span></div>
<div class="line"><a name="l00135"></a><span class="lineno">  135</span>&#160;  <span class="keywordtype">bool</span> SwapCols( <span class="keywordtype">int</span>, <span class="keywordtype">int</span> ); <span class="comment">// ints are col indices to swap</span></div>
<div class="line"><a name="l00136"></a><span class="lineno">  136</span>&#160;  <span class="keywordtype">bool</span> Invert( </div>
<div class="line"><a name="l00137"></a><span class="lineno">  137</span>&#160;          <span class="keywordtype">double</span> <span class="comment">// zero tolerance</span></div>
<div class="line"><a name="l00138"></a><span class="lineno">  138</span>&#160;          );</div>
<div class="line"><a name="l00139"></a><span class="lineno">  139</span>&#160; </div>
<div class="line"><a name="l00140"></a><span class="lineno">  140</span>&#160;  <span class="comment">/*</span></div>
<div class="line"><a name="l00141"></a><span class="lineno">  141</span>&#160;<span class="comment">  Description:</span></div>
<div class="line"><a name="l00142"></a><span class="lineno">  142</span>&#160;<span class="comment">    Set this = A*B.</span></div>
<div class="line"><a name="l00143"></a><span class="lineno">  143</span>&#160;<span class="comment">  Parameters:</span></div>
<div class="line"><a name="l00144"></a><span class="lineno">  144</span>&#160;<span class="comment">    A - [in]</span></div>
<div class="line"><a name="l00145"></a><span class="lineno">  145</span>&#160;<span class="comment">      (Can be this)</span></div>
<div class="line"><a name="l00146"></a><span class="lineno">  146</span>&#160;<span class="comment">    B - [in]</span></div>
<div class="line"><a name="l00147"></a><span class="lineno">  147</span>&#160;<span class="comment">      (Can be this)</span></div>
<div class="line"><a name="l00148"></a><span class="lineno">  148</span>&#160;<span class="comment">  Returns:</span></div>
<div class="line"><a name="l00149"></a><span class="lineno">  149</span>&#160;<span class="comment">    True when A is an mXk matrix and B is a k X n matrix; in which case</span></div>
<div class="line"><a name="l00150"></a><span class="lineno">  150</span>&#160;<span class="comment">    &quot;this&quot; will be an mXn matrix = A*B.</span></div>
<div class="line"><a name="l00151"></a><span class="lineno">  151</span>&#160;<span class="comment">    False when A.ColCount() != B.RowCount().</span></div>
<div class="line"><a name="l00152"></a><span class="lineno">  152</span>&#160;<span class="comment">  */</span></div>
<div class="line"><a name="l00153"></a><span class="lineno">  153</span>&#160;  <span class="keywordtype">bool</span> Multiply( <span class="keyword">const</span> <a class="code" href="class_o_n___matrix.html">ON_Matrix</a>&amp; A, <span class="keyword">const</span> <a class="code" href="class_o_n___matrix.html">ON_Matrix</a>&amp; B );</div>
<div class="line"><a name="l00154"></a><span class="lineno">  154</span>&#160; </div>
<div class="line"><a name="l00155"></a><span class="lineno">  155</span>&#160;  <span class="comment">/*</span></div>
<div class="line"><a name="l00156"></a><span class="lineno">  156</span>&#160;<span class="comment">  Description:</span></div>
<div class="line"><a name="l00157"></a><span class="lineno">  157</span>&#160;<span class="comment">    Set this = A+B.</span></div>
<div class="line"><a name="l00158"></a><span class="lineno">  158</span>&#160;<span class="comment">  Parameters:</span></div>
<div class="line"><a name="l00159"></a><span class="lineno">  159</span>&#160;<span class="comment">    A - [in]</span></div>
<div class="line"><a name="l00160"></a><span class="lineno">  160</span>&#160;<span class="comment">      (Can be this)</span></div>
<div class="line"><a name="l00161"></a><span class="lineno">  161</span>&#160;<span class="comment">    B - [in]</span></div>
<div class="line"><a name="l00162"></a><span class="lineno">  162</span>&#160;<span class="comment">      (Can be this)</span></div>
<div class="line"><a name="l00163"></a><span class="lineno">  163</span>&#160;<span class="comment">  Returns:</span></div>
<div class="line"><a name="l00164"></a><span class="lineno">  164</span>&#160;<span class="comment">    True when A and B are mXn matrices; in which case</span></div>
<div class="line"><a name="l00165"></a><span class="lineno">  165</span>&#160;<span class="comment">    &quot;this&quot; will be an mXn matrix = A+B.</span></div>
<div class="line"><a name="l00166"></a><span class="lineno">  166</span>&#160;<span class="comment">    False when A and B have different sizes.</span></div>
<div class="line"><a name="l00167"></a><span class="lineno">  167</span>&#160;<span class="comment">  */</span></div>
<div class="line"><a name="l00168"></a><span class="lineno">  168</span>&#160;  <span class="keywordtype">bool</span> Add( <span class="keyword">const</span> <a class="code" href="class_o_n___matrix.html">ON_Matrix</a>&amp; A, <span class="keyword">const</span> <a class="code" href="class_o_n___matrix.html">ON_Matrix</a>&amp; B );</div>
<div class="line"><a name="l00169"></a><span class="lineno">  169</span>&#160; </div>
<div class="line"><a name="l00170"></a><span class="lineno">  170</span>&#160; </div>
<div class="line"><a name="l00171"></a><span class="lineno">  171</span>&#160;  <span class="comment">/*</span></div>
<div class="line"><a name="l00172"></a><span class="lineno">  172</span>&#160;<span class="comment">  Description:</span></div>
<div class="line"><a name="l00173"></a><span class="lineno">  173</span>&#160;<span class="comment">    Set this = s*this.</span></div>
<div class="line"><a name="l00174"></a><span class="lineno">  174</span>&#160;<span class="comment">  Parameters:</span></div>
<div class="line"><a name="l00175"></a><span class="lineno">  175</span>&#160;<span class="comment">    s - [in]</span></div>
<div class="line"><a name="l00176"></a><span class="lineno">  176</span>&#160;<span class="comment">  Returns:</span></div>
<div class="line"><a name="l00177"></a><span class="lineno">  177</span>&#160;<span class="comment">    True when A and s are valid.</span></div>
<div class="line"><a name="l00178"></a><span class="lineno">  178</span>&#160;<span class="comment">  */</span></div>
<div class="line"><a name="l00179"></a><span class="lineno">  179</span>&#160;  <span class="keywordtype">bool</span> Scale( <span class="keywordtype">double</span> s );</div>
<div class="line"><a name="l00180"></a><span class="lineno">  180</span>&#160; </div>
<div class="line"><a name="l00181"></a><span class="lineno">  181</span>&#160; </div>
<div class="line"><a name="l00182"></a><span class="lineno">  182</span>&#160;  <span class="comment">// Description:</span></div>
<div class="line"><a name="l00183"></a><span class="lineno">  183</span>&#160;  <span class="comment">//   Row reduce a matrix to calculate rank and determinant.</span></div>
<div class="line"><a name="l00184"></a><span class="lineno">  184</span>&#160;  <span class="comment">// Parameters:</span></div>
<div class="line"><a name="l00185"></a><span class="lineno">  185</span>&#160;  <span class="comment">//   zero_tolerance - [in] (&gt;=0.0) zero tolerance for pivot test</span></div>
<div class="line"><a name="l00186"></a><span class="lineno">  186</span>&#160;  <span class="comment">//       If the absolute value of a pivot is &lt;= zero_tolerance,</span></div>
<div class="line"><a name="l00187"></a><span class="lineno">  187</span>&#160;  <span class="comment">//       then the pivot is assumed to be zero.</span></div>
<div class="line"><a name="l00188"></a><span class="lineno">  188</span>&#160;  <span class="comment">//   determinant - [out] value of determinant is returned here.</span></div>
<div class="line"><a name="l00189"></a><span class="lineno">  189</span>&#160;  <span class="comment">//   pivot - [out] value of the smallest pivot is returned here</span></div>
<div class="line"><a name="l00190"></a><span class="lineno">  190</span>&#160;  <span class="comment">// Returns:</span></div>
<div class="line"><a name="l00191"></a><span class="lineno">  191</span>&#160;  <span class="comment">//   Rank of the matrix.</span></div>
<div class="line"><a name="l00192"></a><span class="lineno">  192</span>&#160;  <span class="comment">// Remarks:</span></div>
<div class="line"><a name="l00193"></a><span class="lineno">  193</span>&#160;  <span class="comment">//   The matrix itself is row reduced so that the result is</span></div>
<div class="line"><a name="l00194"></a><span class="lineno">  194</span>&#160;  <span class="comment">//   an upper triangular matrix with 1&#39;s on the diagonal.</span></div>
<div class="line"><a name="l00195"></a><span class="lineno">  195</span>&#160;  <span class="keywordtype">int</span> RowReduce( <span class="comment">// returns rank</span></div>
<div class="line"><a name="l00196"></a><span class="lineno">  196</span>&#160;    <span class="keywordtype">double</span>,  <span class="comment">// zero_tolerance</span></div>
<div class="line"><a name="l00197"></a><span class="lineno">  197</span>&#160;    <span class="keywordtype">double</span>&amp;, <span class="comment">// determinant</span></div>
<div class="line"><a name="l00198"></a><span class="lineno">  198</span>&#160;    <span class="keywordtype">double</span>&amp;  <span class="comment">// pivot</span></div>
<div class="line"><a name="l00199"></a><span class="lineno">  199</span>&#160;    ); </div>
<div class="line"><a name="l00200"></a><span class="lineno">  200</span>&#160; </div>
<div class="line"><a name="l00201"></a><span class="lineno">  201</span>&#160;  <span class="comment">// Description:</span></div>
<div class="line"><a name="l00202"></a><span class="lineno">  202</span>&#160;  <span class="comment">//   Row reduce a matrix as the first step in solving M*X=B where</span></div>
<div class="line"><a name="l00203"></a><span class="lineno">  203</span>&#160;  <span class="comment">//   B is a column of values.</span></div>
<div class="line"><a name="l00204"></a><span class="lineno">  204</span>&#160;  <span class="comment">// Parameters:</span></div>
<div class="line"><a name="l00205"></a><span class="lineno">  205</span>&#160;  <span class="comment">//   zero_tolerance - [in] (&gt;=0.0) zero tolerance for pivot test</span></div>
<div class="line"><a name="l00206"></a><span class="lineno">  206</span>&#160;  <span class="comment">//       If the absolute value of a pivot is &lt;= zero_tolerance,</span></div>
<div class="line"><a name="l00207"></a><span class="lineno">  207</span>&#160;  <span class="comment">//       then the pivot is assumed to be zero.</span></div>
<div class="line"><a name="l00208"></a><span class="lineno">  208</span>&#160;  <span class="comment">//   B - [in/out] an array of m_row_count values that is row reduced</span></div>
<div class="line"><a name="l00209"></a><span class="lineno">  209</span>&#160;  <span class="comment">//       with the matrix.</span></div>
<div class="line"><a name="l00210"></a><span class="lineno">  210</span>&#160;  <span class="comment">//   determinant - [out] value of determinant is returned here.</span></div>
<div class="line"><a name="l00211"></a><span class="lineno">  211</span>&#160;  <span class="comment">//   pivot - [out] If not NULL, then the value of the smallest </span></div>
<div class="line"><a name="l00212"></a><span class="lineno">  212</span>&#160;  <span class="comment">//       pivot is returned here</span></div>
<div class="line"><a name="l00213"></a><span class="lineno">  213</span>&#160;  <span class="comment">// Returns:</span></div>
<div class="line"><a name="l00214"></a><span class="lineno">  214</span>&#160;  <span class="comment">//   Rank of the matrix.</span></div>
<div class="line"><a name="l00215"></a><span class="lineno">  215</span>&#160;  <span class="comment">// Remarks:</span></div>
<div class="line"><a name="l00216"></a><span class="lineno">  216</span>&#160;  <span class="comment">//   The matrix itself is row reduced so that the result is</span></div>
<div class="line"><a name="l00217"></a><span class="lineno">  217</span>&#160;  <span class="comment">//   an upper triangular matrix with 1&#39;s on the diagonal.</span></div>
<div class="line"><a name="l00218"></a><span class="lineno">  218</span>&#160;  <span class="comment">// Example:</span></div>
<div class="line"><a name="l00219"></a><span class="lineno">  219</span>&#160;  <span class="comment">//   Solve M*X=B;</span></div>
<div class="line"><a name="l00220"></a><span class="lineno">  220</span>&#160;  <span class="comment">//   double B[m] = ...;</span></div>
<div class="line"><a name="l00221"></a><span class="lineno">  221</span>&#160;  <span class="comment">//   double B[n] = ...;</span></div>
<div class="line"><a name="l00222"></a><span class="lineno">  222</span>&#160;  <span class="comment">//   ON_Matrix M(m,n) = ...;</span></div>
<div class="line"><a name="l00223"></a><span class="lineno">  223</span>&#160;  <span class="comment">//   M.RowReduce(ON_ZERO_TOLERANCE,B); // modifies M and B</span></div>
<div class="line"><a name="l00224"></a><span class="lineno">  224</span>&#160;  <span class="comment">//   M.BackSolve(m,B,X); // solution is in X</span></div>
<div class="line"><a name="l00225"></a><span class="lineno">  225</span>&#160;  <span class="comment">// See Also: </span></div>
<div class="line"><a name="l00226"></a><span class="lineno">  226</span>&#160;  <span class="comment">//   ON_Matrix::BackSolve</span></div>
<div class="line"><a name="l00227"></a><span class="lineno">  227</span>&#160;  <span class="keywordtype">int</span> RowReduce(</div>
<div class="line"><a name="l00228"></a><span class="lineno">  228</span>&#160;    <span class="keywordtype">double</span>,        <span class="comment">// zero_tolerance</span></div>
<div class="line"><a name="l00229"></a><span class="lineno">  229</span>&#160;    <span class="keywordtype">double</span>*,       <span class="comment">// B</span></div>
<div class="line"><a name="l00230"></a><span class="lineno">  230</span>&#160;    <span class="keywordtype">double</span>* = NULL <span class="comment">// pivot</span></div>
<div class="line"><a name="l00231"></a><span class="lineno">  231</span>&#160;    ); </div>
<div class="line"><a name="l00232"></a><span class="lineno">  232</span>&#160; </div>
<div class="line"><a name="l00233"></a><span class="lineno">  233</span>&#160;  <span class="comment">// Description:</span></div>
<div class="line"><a name="l00234"></a><span class="lineno">  234</span>&#160;  <span class="comment">//   Row reduce a matrix as the first step in solving M*X=B where</span></div>
<div class="line"><a name="l00235"></a><span class="lineno">  235</span>&#160;  <span class="comment">//   B is a column of 3d points</span></div>
<div class="line"><a name="l00236"></a><span class="lineno">  236</span>&#160;  <span class="comment">// Parameters:</span></div>
<div class="line"><a name="l00237"></a><span class="lineno">  237</span>&#160;  <span class="comment">//   zero_tolerance - [in] (&gt;=0.0) zero tolerance for pivot test</span></div>
<div class="line"><a name="l00238"></a><span class="lineno">  238</span>&#160;  <span class="comment">//       If the absolute value of a pivot is &lt;= zero_tolerance,</span></div>
<div class="line"><a name="l00239"></a><span class="lineno">  239</span>&#160;  <span class="comment">//       then the pivot is assumed to be zero.</span></div>
<div class="line"><a name="l00240"></a><span class="lineno">  240</span>&#160;  <span class="comment">//   B - [in/out] an array of m_row_count 3d points that is </span></div>
<div class="line"><a name="l00241"></a><span class="lineno">  241</span>&#160;  <span class="comment">//       row reduced with the matrix.</span></div>
<div class="line"><a name="l00242"></a><span class="lineno">  242</span>&#160;  <span class="comment">//   determinant - [out] value of determinant is returned here.</span></div>
<div class="line"><a name="l00243"></a><span class="lineno">  243</span>&#160;  <span class="comment">//   pivot - [out] If not NULL, then the value of the smallest </span></div>
<div class="line"><a name="l00244"></a><span class="lineno">  244</span>&#160;  <span class="comment">//       pivot is returned here</span></div>
<div class="line"><a name="l00245"></a><span class="lineno">  245</span>&#160;  <span class="comment">// Returns:</span></div>
<div class="line"><a name="l00246"></a><span class="lineno">  246</span>&#160;  <span class="comment">//   Rank of the matrix.</span></div>
<div class="line"><a name="l00247"></a><span class="lineno">  247</span>&#160;  <span class="comment">// Remarks:</span></div>
<div class="line"><a name="l00248"></a><span class="lineno">  248</span>&#160;  <span class="comment">//   The matrix itself is row reduced so that the result is</span></div>
<div class="line"><a name="l00249"></a><span class="lineno">  249</span>&#160;  <span class="comment">//   an upper triangular matrix with 1&#39;s on the diagonal.</span></div>
<div class="line"><a name="l00250"></a><span class="lineno">  250</span>&#160;  <span class="comment">// See Also: </span></div>
<div class="line"><a name="l00251"></a><span class="lineno">  251</span>&#160;  <span class="comment">//   ON_Matrix::BackSolve</span></div>
<div class="line"><a name="l00252"></a><span class="lineno">  252</span>&#160;  <span class="keywordtype">int</span> RowReduce(</div>
<div class="line"><a name="l00253"></a><span class="lineno">  253</span>&#160;    <span class="keywordtype">double</span>,        <span class="comment">// zero_tolerance</span></div>
<div class="line"><a name="l00254"></a><span class="lineno">  254</span>&#160;    <a class="code" href="class_o_n__3d_point.html">ON_3dPoint</a>*,   <span class="comment">// B</span></div>
<div class="line"><a name="l00255"></a><span class="lineno">  255</span>&#160;    <span class="keywordtype">double</span>* = NULL <span class="comment">// pivot</span></div>
<div class="line"><a name="l00256"></a><span class="lineno">  256</span>&#160;    ); </div>
<div class="line"><a name="l00257"></a><span class="lineno">  257</span>&#160; </div>
<div class="line"><a name="l00258"></a><span class="lineno">  258</span>&#160;  <span class="comment">// Description:</span></div>
<div class="line"><a name="l00259"></a><span class="lineno">  259</span>&#160;  <span class="comment">//   Row reduce a matrix as the first step in solving M*X=B where</span></div>
<div class="line"><a name="l00260"></a><span class="lineno">  260</span>&#160;  <span class="comment">//   B is a column arbitrary dimension points.</span></div>
<div class="line"><a name="l00261"></a><span class="lineno">  261</span>&#160;  <span class="comment">// Parameters:</span></div>
<div class="line"><a name="l00262"></a><span class="lineno">  262</span>&#160;  <span class="comment">//   zero_tolerance - [in] (&gt;=0.0) zero tolerance for pivot test</span></div>
<div class="line"><a name="l00263"></a><span class="lineno">  263</span>&#160;  <span class="comment">//       If a the absolute value of a pivot is &lt;= zero_tolerance,</span></div>
<div class="line"><a name="l00264"></a><span class="lineno">  264</span>&#160;  <span class="comment">//       then the pivoit is assumed to be zero.</span></div>
<div class="line"><a name="l00265"></a><span class="lineno">  265</span>&#160;  <span class="comment">//   pt_dim - [in] dimension of points</span></div>
<div class="line"><a name="l00266"></a><span class="lineno">  266</span>&#160;  <span class="comment">//   pt_stride - [in] stride between points (&gt;=pt_dim)</span></div>
<div class="line"><a name="l00267"></a><span class="lineno">  267</span>&#160;  <span class="comment">//   pt - [in/out] array of m_row_count*pt_stride values.</span></div>
<div class="line"><a name="l00268"></a><span class="lineno">  268</span>&#160;  <span class="comment">//        The i-th point is</span></div>
<div class="line"><a name="l00269"></a><span class="lineno">  269</span>&#160;  <span class="comment">//        (pt[i*pt_stride],...,pt[i*pt_stride+pt_dim-1]).</span></div>
<div class="line"><a name="l00270"></a><span class="lineno">  270</span>&#160;  <span class="comment">//        This array of points is row reduced along with the </span></div>
<div class="line"><a name="l00271"></a><span class="lineno">  271</span>&#160;  <span class="comment">//        matrix.</span></div>
<div class="line"><a name="l00272"></a><span class="lineno">  272</span>&#160;  <span class="comment">//   pivot - [out] If not NULL, then the value of the smallest </span></div>
<div class="line"><a name="l00273"></a><span class="lineno">  273</span>&#160;  <span class="comment">//       pivot is returned here</span></div>
<div class="line"><a name="l00274"></a><span class="lineno">  274</span>&#160;  <span class="comment">// Returns:</span></div>
<div class="line"><a name="l00275"></a><span class="lineno">  275</span>&#160;  <span class="comment">//   Rank of the matrix.</span></div>
<div class="line"><a name="l00276"></a><span class="lineno">  276</span>&#160;  <span class="comment">// Remarks:</span></div>
<div class="line"><a name="l00277"></a><span class="lineno">  277</span>&#160;  <span class="comment">//   The matrix itself is row reduced so that the result is</span></div>
<div class="line"><a name="l00278"></a><span class="lineno">  278</span>&#160;  <span class="comment">//   an upper triangular matrix with 1&#39;s on the diagonal.</span></div>
<div class="line"><a name="l00279"></a><span class="lineno">  279</span>&#160;  <span class="comment">// See Also: </span></div>
<div class="line"><a name="l00280"></a><span class="lineno">  280</span>&#160;  <span class="comment">//   ON_Matrix::BackSolve</span></div>
<div class="line"><a name="l00281"></a><span class="lineno">  281</span>&#160;  <span class="keywordtype">int</span> RowReduce( <span class="comment">// returns rank</span></div>
<div class="line"><a name="l00282"></a><span class="lineno">  282</span>&#160;    <span class="keywordtype">double</span>,      <span class="comment">// zero_tolerance</span></div>
<div class="line"><a name="l00283"></a><span class="lineno">  283</span>&#160;    <span class="keywordtype">int</span>,         <span class="comment">// pt_dim</span></div>
<div class="line"><a name="l00284"></a><span class="lineno">  284</span>&#160;    <span class="keywordtype">int</span>,         <span class="comment">// pt_stride</span></div>
<div class="line"><a name="l00285"></a><span class="lineno">  285</span>&#160;    <span class="keywordtype">double</span>*,     <span class="comment">// pt</span></div>
<div class="line"><a name="l00286"></a><span class="lineno">  286</span>&#160;    <span class="keywordtype">double</span>* = NULL <span class="comment">// pivot</span></div>
<div class="line"><a name="l00287"></a><span class="lineno">  287</span>&#160;    ); </div>
<div class="line"><a name="l00288"></a><span class="lineno">  288</span>&#160; </div>
<div class="line"><a name="l00289"></a><span class="lineno">  289</span>&#160;  <span class="comment">// Description:</span></div>
<div class="line"><a name="l00290"></a><span class="lineno">  290</span>&#160;  <span class="comment">//   Solve M*X=B where M is upper triangular with a unit diagonal and</span></div>
<div class="line"><a name="l00291"></a><span class="lineno">  291</span>&#160;  <span class="comment">//   B is a column of values.</span></div>
<div class="line"><a name="l00292"></a><span class="lineno">  292</span>&#160;  <span class="comment">// Parameters:</span></div>
<div class="line"><a name="l00293"></a><span class="lineno">  293</span>&#160;  <span class="comment">//   zero_tolerance - [in] (&gt;=0.0) used to test for &quot;zero&quot; values in B</span></div>
<div class="line"><a name="l00294"></a><span class="lineno">  294</span>&#160;  <span class="comment">//       in under determined systems of equations.</span></div>
<div class="line"><a name="l00295"></a><span class="lineno">  295</span>&#160;  <span class="comment">//   Bsize - [in] (&gt;=m_row_count) length of B.  The values in</span></div>
<div class="line"><a name="l00296"></a><span class="lineno">  296</span>&#160;  <span class="comment">//       B[m_row_count],...,B[Bsize-1] are tested to make sure they are</span></div>
<div class="line"><a name="l00297"></a><span class="lineno">  297</span>&#160;  <span class="comment">//       &quot;zero&quot;.</span></div>
<div class="line"><a name="l00298"></a><span class="lineno">  298</span>&#160;  <span class="comment">//   B - [in] array of length Bsize.</span></div>
<div class="line"><a name="l00299"></a><span class="lineno">  299</span>&#160;  <span class="comment">//   X - [out] array of length m_col_count.  Solutions returned here.</span></div>
<div class="line"><a name="l00300"></a><span class="lineno">  300</span>&#160;  <span class="comment">// Remarks:</span></div>
<div class="line"><a name="l00301"></a><span class="lineno">  301</span>&#160;  <span class="comment">//   Actual values M[i][j] with i &lt;= j are ignored. </span></div>
<div class="line"><a name="l00302"></a><span class="lineno">  302</span>&#160;  <span class="comment">//   M[i][i] is assumed to be one and M[i][j] i&lt;j is assumed to be zero.</span></div>
<div class="line"><a name="l00303"></a><span class="lineno">  303</span>&#160;  <span class="comment">//   For square M, B and X can point to the same memory.</span></div>
<div class="line"><a name="l00304"></a><span class="lineno">  304</span>&#160;  <span class="comment">// See Also:</span></div>
<div class="line"><a name="l00305"></a><span class="lineno">  305</span>&#160;  <span class="comment">//   ON_Matrix::RowReduce</span></div>
<div class="line"><a name="l00306"></a><span class="lineno">  306</span>&#160;  <span class="keywordtype">bool</span> BackSolve(</div>
<div class="line"><a name="l00307"></a><span class="lineno">  307</span>&#160;    <span class="keywordtype">double</span>,        <span class="comment">// zero_tolerance</span></div>
<div class="line"><a name="l00308"></a><span class="lineno">  308</span>&#160;    <span class="keywordtype">int</span>,           <span class="comment">// Bsize</span></div>
<div class="line"><a name="l00309"></a><span class="lineno">  309</span>&#160;    <span class="keyword">const</span> <span class="keywordtype">double</span>*, <span class="comment">// B</span></div>
<div class="line"><a name="l00310"></a><span class="lineno">  310</span>&#160;    <span class="keywordtype">double</span>*        <span class="comment">// X</span></div>
<div class="line"><a name="l00311"></a><span class="lineno">  311</span>&#160;      ) <span class="keyword">const</span>;</div>
<div class="line"><a name="l00312"></a><span class="lineno">  312</span>&#160; </div>
<div class="line"><a name="l00313"></a><span class="lineno">  313</span>&#160;  <span class="comment">// Description:</span></div>
<div class="line"><a name="l00314"></a><span class="lineno">  314</span>&#160;  <span class="comment">//   Solve M*X=B where M is upper triangular with a unit diagonal and</span></div>
<div class="line"><a name="l00315"></a><span class="lineno">  315</span>&#160;  <span class="comment">//   B is a column of 3d points.</span></div>
<div class="line"><a name="l00316"></a><span class="lineno">  316</span>&#160;  <span class="comment">// Parameters:</span></div>
<div class="line"><a name="l00317"></a><span class="lineno">  317</span>&#160;  <span class="comment">//   zero_tolerance - [in] (&gt;=0.0) used to test for &quot;zero&quot; values in B</span></div>
<div class="line"><a name="l00318"></a><span class="lineno">  318</span>&#160;  <span class="comment">//       in under determined systems of equations.</span></div>
<div class="line"><a name="l00319"></a><span class="lineno">  319</span>&#160;  <span class="comment">//   Bsize - [in] (&gt;=m_row_count) length of B.  The values in</span></div>
<div class="line"><a name="l00320"></a><span class="lineno">  320</span>&#160;  <span class="comment">//       B[m_row_count],...,B[Bsize-1] are tested to make sure they are</span></div>
<div class="line"><a name="l00321"></a><span class="lineno">  321</span>&#160;  <span class="comment">//       &quot;zero&quot;.</span></div>
<div class="line"><a name="l00322"></a><span class="lineno">  322</span>&#160;  <span class="comment">//   B - [in] array of length Bsize.</span></div>
<div class="line"><a name="l00323"></a><span class="lineno">  323</span>&#160;  <span class="comment">//   X - [out] array of length m_col_count.  Solutions returned here.</span></div>
<div class="line"><a name="l00324"></a><span class="lineno">  324</span>&#160;  <span class="comment">// Remarks:</span></div>
<div class="line"><a name="l00325"></a><span class="lineno">  325</span>&#160;  <span class="comment">//   Actual values M[i][j] with i &lt;= j are ignored. </span></div>
<div class="line"><a name="l00326"></a><span class="lineno">  326</span>&#160;  <span class="comment">//   M[i][i] is assumed to be one and M[i][j] i&lt;j is assumed to be zero.</span></div>
<div class="line"><a name="l00327"></a><span class="lineno">  327</span>&#160;  <span class="comment">//   For square M, B and X can point to the same memory.</span></div>
<div class="line"><a name="l00328"></a><span class="lineno">  328</span>&#160;  <span class="comment">// See Also:</span></div>
<div class="line"><a name="l00329"></a><span class="lineno">  329</span>&#160;  <span class="comment">//   ON_Matrix::RowReduce</span></div>
<div class="line"><a name="l00330"></a><span class="lineno">  330</span>&#160;  <span class="keywordtype">bool</span> BackSolve(</div>
<div class="line"><a name="l00331"></a><span class="lineno">  331</span>&#160;    <span class="keywordtype">double</span>,            <span class="comment">// zero_tolerance</span></div>
<div class="line"><a name="l00332"></a><span class="lineno">  332</span>&#160;    <span class="keywordtype">int</span>,               <span class="comment">// Bsize</span></div>
<div class="line"><a name="l00333"></a><span class="lineno">  333</span>&#160;    <span class="keyword">const</span> <a class="code" href="class_o_n__3d_point.html">ON_3dPoint</a>*, <span class="comment">// B</span></div>
<div class="line"><a name="l00334"></a><span class="lineno">  334</span>&#160;    <a class="code" href="class_o_n__3d_point.html">ON_3dPoint</a>*        <span class="comment">// X</span></div>
<div class="line"><a name="l00335"></a><span class="lineno">  335</span>&#160;      ) <span class="keyword">const</span>;</div>
<div class="line"><a name="l00336"></a><span class="lineno">  336</span>&#160; </div>
<div class="line"><a name="l00337"></a><span class="lineno">  337</span>&#160;  <span class="comment">// Description:</span></div>
<div class="line"><a name="l00338"></a><span class="lineno">  338</span>&#160;  <span class="comment">//   Solve M*X=B where M is upper triangular with a unit diagonal and</span></div>
<div class="line"><a name="l00339"></a><span class="lineno">  339</span>&#160;  <span class="comment">//   B is a column of points</span></div>
<div class="line"><a name="l00340"></a><span class="lineno">  340</span>&#160;  <span class="comment">// Parameters:</span></div>
<div class="line"><a name="l00341"></a><span class="lineno">  341</span>&#160;  <span class="comment">//   zero_tolerance - [in] (&gt;=0.0) used to test for &quot;zero&quot; values in B</span></div>
<div class="line"><a name="l00342"></a><span class="lineno">  342</span>&#160;  <span class="comment">//       in under determined systems of equations.</span></div>
<div class="line"><a name="l00343"></a><span class="lineno">  343</span>&#160;  <span class="comment">//   pt_dim - [in] dimension of points</span></div>
<div class="line"><a name="l00344"></a><span class="lineno">  344</span>&#160;  <span class="comment">//   Bsize - [in] (&gt;=m_row_count) number of points in B[].  The points</span></div>
<div class="line"><a name="l00345"></a><span class="lineno">  345</span>&#160;  <span class="comment">//       correspoinding to indices m_row_count, ..., (Bsize-1)</span></div>
<div class="line"><a name="l00346"></a><span class="lineno">  346</span>&#160;  <span class="comment">//       are tested to make sure they are &quot;zero&quot;.</span></div>
<div class="line"><a name="l00347"></a><span class="lineno">  347</span>&#160;  <span class="comment">//   Bpt_stride - [in] stride between B points (&gt;=pt_dim)</span></div>
<div class="line"><a name="l00348"></a><span class="lineno">  348</span>&#160;  <span class="comment">//   Bpt - [in/out] array of m_row_count*Bpt_stride values.</span></div>
<div class="line"><a name="l00349"></a><span class="lineno">  349</span>&#160;  <span class="comment">//        The i-th B point is</span></div>
<div class="line"><a name="l00350"></a><span class="lineno">  350</span>&#160;  <span class="comment">//        (Bpt[i*Bpt_stride],...,Bpt[i*Bpt_stride+pt_dim-1]).</span></div>
<div class="line"><a name="l00351"></a><span class="lineno">  351</span>&#160;  <span class="comment">//   Xpt_stride - [in] stride between X points (&gt;=pt_dim)</span></div>
<div class="line"><a name="l00352"></a><span class="lineno">  352</span>&#160;  <span class="comment">//   Xpt - [out] array of m_col_count*Xpt_stride values.</span></div>
<div class="line"><a name="l00353"></a><span class="lineno">  353</span>&#160;  <span class="comment">//        The i-th X point is</span></div>
<div class="line"><a name="l00354"></a><span class="lineno">  354</span>&#160;  <span class="comment">//        (Xpt[i*Xpt_stride],...,Xpt[i*Xpt_stride+pt_dim-1]).</span></div>
<div class="line"><a name="l00355"></a><span class="lineno">  355</span>&#160;  <span class="comment">// Remarks:</span></div>
<div class="line"><a name="l00356"></a><span class="lineno">  356</span>&#160;  <span class="comment">//   Actual values M[i][j] with i &lt;= j are ignored. </span></div>
<div class="line"><a name="l00357"></a><span class="lineno">  357</span>&#160;  <span class="comment">//   M[i][i] is assumed to be one and M[i][j] i&lt;j is assumed to be zero.</span></div>
<div class="line"><a name="l00358"></a><span class="lineno">  358</span>&#160;  <span class="comment">//   For square M, B and X can point to the same memory.</span></div>
<div class="line"><a name="l00359"></a><span class="lineno">  359</span>&#160;  <span class="comment">// See Also:</span></div>
<div class="line"><a name="l00360"></a><span class="lineno">  360</span>&#160;  <span class="comment">//   ON_Matrix::RowReduce</span></div>
<div class="line"><a name="l00361"></a><span class="lineno">  361</span>&#160;  <span class="keywordtype">bool</span> BackSolve(</div>
<div class="line"><a name="l00362"></a><span class="lineno">  362</span>&#160;    <span class="keywordtype">double</span>,       <span class="comment">// zero_tolerance</span></div>
<div class="line"><a name="l00363"></a><span class="lineno">  363</span>&#160;    <span class="keywordtype">int</span>,          <span class="comment">// pt_dim</span></div>
<div class="line"><a name="l00364"></a><span class="lineno">  364</span>&#160;    <span class="keywordtype">int</span>,          <span class="comment">// Bsize</span></div>
<div class="line"><a name="l00365"></a><span class="lineno">  365</span>&#160;    <span class="keywordtype">int</span>,          <span class="comment">// Bpt_stride</span></div>
<div class="line"><a name="l00366"></a><span class="lineno">  366</span>&#160;    <span class="keyword">const</span> <span class="keywordtype">double</span>*,<span class="comment">// Bpt</span></div>
<div class="line"><a name="l00367"></a><span class="lineno">  367</span>&#160;    <span class="keywordtype">int</span>,          <span class="comment">// Xpt_stride</span></div>
<div class="line"><a name="l00368"></a><span class="lineno">  368</span>&#160;    <span class="keywordtype">double</span>*       <span class="comment">// Xpt</span></div>
<div class="line"><a name="l00369"></a><span class="lineno">  369</span>&#160;      ) <span class="keyword">const</span>;</div>
<div class="line"><a name="l00370"></a><span class="lineno">  370</span>&#160; </div>
<div class="line"><a name="l00371"></a><span class="lineno">  371</span>&#160;  <span class="keywordtype">bool</span> IsRowOrthoganal() <span class="keyword">const</span>;</div>
<div class="line"><a name="l00372"></a><span class="lineno">  372</span>&#160;  <span class="keywordtype">bool</span> IsRowOrthoNormal() <span class="keyword">const</span>;</div>
<div class="line"><a name="l00373"></a><span class="lineno">  373</span>&#160; </div>
<div class="line"><a name="l00374"></a><span class="lineno">  374</span>&#160;  <span class="keywordtype">bool</span> IsColOrthoganal() <span class="keyword">const</span>;</div>
<div class="line"><a name="l00375"></a><span class="lineno">  375</span>&#160;  <span class="keywordtype">bool</span> IsColOrthoNormal() <span class="keyword">const</span>;</div>
<div class="line"><a name="l00376"></a><span class="lineno">  376</span>&#160; </div>
<div class="line"><a name="l00377"></a><span class="lineno">  377</span>&#160; </div>
<div class="line"><a name="l00378"></a><span class="lineno">  378</span>&#160;  <span class="keywordtype">double</span>** m; <span class="comment">// m[i][j] = value at row i and column j</span></div>
<div class="line"><a name="l00379"></a><span class="lineno">  379</span>&#160;              <span class="comment">//           0 &lt;= i &lt; RowCount()</span></div>
<div class="line"><a name="l00380"></a><span class="lineno">  380</span>&#160;              <span class="comment">//           0 &lt;= j &lt; ColCount()</span></div>
<div class="line"><a name="l00381"></a><span class="lineno">  381</span>&#160;<span class="keyword">private</span>:</div>
<div class="line"><a name="l00382"></a><span class="lineno">  382</span>&#160;  <span class="keywordtype">int</span> m_row_count;</div>
<div class="line"><a name="l00383"></a><span class="lineno">  383</span>&#160;  <span class="keywordtype">int</span> m_col_count;</div>
<div class="line"><a name="l00384"></a><span class="lineno">  384</span>&#160;  <span class="comment">// m_rowmem[i][j] = row i+m_row_offset and column j+m_col_offset.</span></div>
<div class="line"><a name="l00385"></a><span class="lineno">  385</span>&#160;  <a class="code" href="class_o_n___simple_array.html">ON_SimpleArray&lt;double*&gt;</a> m_rowmem; </div>
<div class="line"><a name="l00386"></a><span class="lineno">  386</span>&#160;    <span class="keywordtype">double</span>** m_Mmem; <span class="comment">// used by Create(row_count,col_count,user_memory,true);</span></div>
<div class="line"><a name="l00387"></a><span class="lineno">  387</span>&#160;    <span class="keywordtype">int</span>   m_row_offset; <span class="comment">// = ri0 when sub-matrix constructor is used</span></div>
<div class="line"><a name="l00388"></a><span class="lineno">  388</span>&#160;    <span class="keywordtype">int</span>   m_col_offset; <span class="comment">// = ci0 when sub-matrix constructor is used</span></div>
<div class="line"><a name="l00389"></a><span class="lineno">  389</span>&#160;  <span class="keywordtype">void</span>* m_cmem;</div>
<div class="line"><a name="l00390"></a><span class="lineno">  390</span>&#160;  <span class="comment">// returns 0 based arrays, even in submatrix case.</span></div>
<div class="line"><a name="l00391"></a><span class="lineno">  391</span>&#160;  <span class="keywordtype">double</span> <span class="keyword">const</span> * <span class="keyword">const</span> * ThisM() <span class="keyword">const</span>;</div>
<div class="line"><a name="l00392"></a><span class="lineno">  392</span>&#160;  <span class="keywordtype">double</span> * * ThisM();</div>
<div class="line"><a name="l00393"></a><span class="lineno">  393</span>&#160;};</div>
<div class="line"><a name="l00394"></a><span class="lineno">  394</span>&#160; </div>
<div class="line"><a name="l00395"></a><span class="lineno">  395</span>&#160;<span class="comment">/*</span></div>
<div class="line"><a name="l00396"></a><span class="lineno">  396</span>&#160;<span class="comment">Description:</span></div>
<div class="line"><a name="l00397"></a><span class="lineno">  397</span>&#160;<span class="comment">  Calculate the singular value decomposition of a matrix.</span></div>
<div class="line"><a name="l00398"></a><span class="lineno">  398</span>&#160;<span class="comment"></span> </div>
<div class="line"><a name="l00399"></a><span class="lineno">  399</span>&#160;<span class="comment">Parameters:</span></div>
<div class="line"><a name="l00400"></a><span class="lineno">  400</span>&#160;<span class="comment">  row_count - [in]</span></div>
<div class="line"><a name="l00401"></a><span class="lineno">  401</span>&#160;<span class="comment">    number of rows in matrix A</span></div>
<div class="line"><a name="l00402"></a><span class="lineno">  402</span>&#160;<span class="comment">  col_count - [in]</span></div>
<div class="line"><a name="l00403"></a><span class="lineno">  403</span>&#160;<span class="comment">    number of columns in matrix A</span></div>
<div class="line"><a name="l00404"></a><span class="lineno">  404</span>&#160;<span class="comment">  A - [in]</span></div>
<div class="line"><a name="l00405"></a><span class="lineno">  405</span>&#160;<span class="comment">    Matrix for which you want the singular value decomposition.</span></div>
<div class="line"><a name="l00406"></a><span class="lineno">  406</span>&#160;<span class="comment">    A[0][0] = coefficeint in the first row and first column.</span></div>
<div class="line"><a name="l00407"></a><span class="lineno">  407</span>&#160;<span class="comment">    A[row_count-1][col_count-1] = coefficeint in the last row</span></div>
<div class="line"><a name="l00408"></a><span class="lineno">  408</span>&#160;<span class="comment">    and last column.</span></div>
<div class="line"><a name="l00409"></a><span class="lineno">  409</span>&#160;<span class="comment">  U - [out]</span></div>
<div class="line"><a name="l00410"></a><span class="lineno">  410</span>&#160;<span class="comment">    The singular value decomposition of A is U*Diag(W)*Transpose(V),</span></div>
<div class="line"><a name="l00411"></a><span class="lineno">  411</span>&#160;<span class="comment">    where U has the same size as A, Diag(W) is a col_count X col_count</span></div>
<div class="line"><a name="l00412"></a><span class="lineno">  412</span>&#160;<span class="comment">    diagonal matrix with (W[0],...,W[col_count-1]) on the diagonal</span></div>
<div class="line"><a name="l00413"></a><span class="lineno">  413</span>&#160;<span class="comment">    and V is a col_count X col_count matrix.</span></div>
<div class="line"><a name="l00414"></a><span class="lineno">  414</span>&#160;<span class="comment">    U and A may be the same pointer.  If the input value of U is</span></div>
<div class="line"><a name="l00415"></a><span class="lineno">  415</span>&#160;<span class="comment">    null, heap storage will be allocated using onmalloc()</span></div>
<div class="line"><a name="l00416"></a><span class="lineno">  416</span>&#160;<span class="comment">    and the calling function must call onfree(U).  If the input</span></div>
<div class="line"><a name="l00417"></a><span class="lineno">  417</span>&#160;<span class="comment">    value of U is not null, U[i] must point to an array of col_count</span></div>
<div class="line"><a name="l00418"></a><span class="lineno">  418</span>&#160;<span class="comment">    doubles.  </span></div>
<div class="line"><a name="l00419"></a><span class="lineno">  419</span>&#160;<span class="comment">  W - [out]</span></div>
<div class="line"><a name="l00420"></a><span class="lineno">  420</span>&#160;<span class="comment">    If the input value W is null, then heap storage will be allocated</span></div>
<div class="line"><a name="l00421"></a><span class="lineno">  421</span>&#160;<span class="comment">    using onmalloc() and the calling function must call onfree(W).</span></div>
<div class="line"><a name="l00422"></a><span class="lineno">  422</span>&#160;<span class="comment">    If the input value of W is not null, then W must point to</span></div>
<div class="line"><a name="l00423"></a><span class="lineno">  423</span>&#160;<span class="comment">    an array of col_count doubles.</span></div>
<div class="line"><a name="l00424"></a><span class="lineno">  424</span>&#160;<span class="comment">  V - [out]</span></div>
<div class="line"><a name="l00425"></a><span class="lineno">  425</span>&#160;<span class="comment">    If the input value V is null, then heap storage will be allocated</span></div>
<div class="line"><a name="l00426"></a><span class="lineno">  426</span>&#160;<span class="comment">    using onmalloc() and the calling function must call onfree(V).</span></div>
<div class="line"><a name="l00427"></a><span class="lineno">  427</span>&#160;<span class="comment">    If the input value of V is not null, then V[i] must point</span></div>
<div class="line"><a name="l00428"></a><span class="lineno">  428</span>&#160;<span class="comment">    to an array of col_count doubles.</span></div>
<div class="line"><a name="l00429"></a><span class="lineno">  429</span>&#160;<span class="comment"></span> </div>
<div class="line"><a name="l00430"></a><span class="lineno">  430</span>&#160;<span class="comment">Example:</span></div>
<div class="line"><a name="l00431"></a><span class="lineno">  431</span>&#160;<span class="comment"></span> </div>
<div class="line"><a name="l00432"></a><span class="lineno">  432</span>&#160;<span class="comment">          int m = row_count;</span></div>
<div class="line"><a name="l00433"></a><span class="lineno">  433</span>&#160;<span class="comment">          int n = col_count;</span></div>
<div class="line"><a name="l00434"></a><span class="lineno">  434</span>&#160;<span class="comment">          ON_Matrix A(m,n);</span></div>
<div class="line"><a name="l00435"></a><span class="lineno">  435</span>&#160;<span class="comment">          for (i = 0; i &lt; m; i++ ) for ( j = 0; j &lt; n; j++ )</span></div>
<div class="line"><a name="l00436"></a><span class="lineno">  436</span>&#160;<span class="comment">          {</span></div>
<div class="line"><a name="l00437"></a><span class="lineno">  437</span>&#160;<span class="comment">            A[i][j] = ...;</span></div>
<div class="line"><a name="l00438"></a><span class="lineno">  438</span>&#160;<span class="comment">          }</span></div>
<div class="line"><a name="l00439"></a><span class="lineno">  439</span>&#160;<span class="comment">          ON_Matrix U(m,n);</span></div>
<div class="line"><a name="l00440"></a><span class="lineno">  440</span>&#160;<span class="comment">          double* W = 0; // ON_GetMatrixSVD() will allocate W</span></div>
<div class="line"><a name="l00441"></a><span class="lineno">  441</span>&#160;<span class="comment">          ON_Matrix V(n,n);</span></div>
<div class="line"><a name="l00442"></a><span class="lineno">  442</span>&#160;<span class="comment">          bool rc = ON_GetMatrixSVD(m,n,A.m,U.m,W,V.m);</span></div>
<div class="line"><a name="l00443"></a><span class="lineno">  443</span>&#160;<span class="comment">          ...</span></div>
<div class="line"><a name="l00444"></a><span class="lineno">  444</span>&#160;<span class="comment">          onfree(W); // W allocated in ON_GetMatrixSVD()</span></div>
<div class="line"><a name="l00445"></a><span class="lineno">  445</span>&#160;<span class="comment"></span> </div>
<div class="line"><a name="l00446"></a><span class="lineno">  446</span>&#160;<span class="comment">Returns:</span></div>
<div class="line"><a name="l00447"></a><span class="lineno">  447</span>&#160;<span class="comment">  True if the singular value decomposition was cacluated.</span></div>
<div class="line"><a name="l00448"></a><span class="lineno">  448</span>&#160;<span class="comment">  False if the algorithm failed to converge.</span></div>
<div class="line"><a name="l00449"></a><span class="lineno">  449</span>&#160;<span class="comment">*/</span></div>
<div class="line"><a name="l00450"></a><span class="lineno">  450</span>&#160;ON_DECL</div>
<div class="line"><a name="l00451"></a><span class="lineno">  451</span>&#160;<span class="keywordtype">bool</span> ON_GetMatrixSVD(</div>
<div class="line"><a name="l00452"></a><span class="lineno">  452</span>&#160;  <span class="keywordtype">int</span> row_count,</div>
<div class="line"><a name="l00453"></a><span class="lineno">  453</span>&#160;  <span class="keywordtype">int</span> col_count,</div>
<div class="line"><a name="l00454"></a><span class="lineno">  454</span>&#160;  <span class="keywordtype">double</span> <span class="keyword">const</span> * <span class="keyword">const</span> * A,</div>
<div class="line"><a name="l00455"></a><span class="lineno">  455</span>&#160;  <span class="keywordtype">double</span>**&amp; U,</div>
<div class="line"><a name="l00456"></a><span class="lineno">  456</span>&#160;  <span class="keywordtype">double</span>*&amp; W,</div>
<div class="line"><a name="l00457"></a><span class="lineno">  457</span>&#160;  <span class="keywordtype">double</span>**&amp; V</div>
<div class="line"><a name="l00458"></a><span class="lineno">  458</span>&#160;  );</div>
<div class="line"><a name="l00459"></a><span class="lineno">  459</span>&#160; </div>
<div class="line"><a name="l00460"></a><span class="lineno">  460</span>&#160;<span class="comment">/*</span></div>
<div class="line"><a name="l00461"></a><span class="lineno">  461</span>&#160;<span class="comment">Description:</span></div>
<div class="line"><a name="l00462"></a><span class="lineno">  462</span>&#160;<span class="comment">  Invert the diagonal matrix in a the singular value decomposition.</span></div>
<div class="line"><a name="l00463"></a><span class="lineno">  463</span>&#160;<span class="comment">Parameters:</span></div>
<div class="line"><a name="l00464"></a><span class="lineno">  464</span>&#160;<span class="comment">  count - [in] number of elements in W</span></div>
<div class="line"><a name="l00465"></a><span class="lineno">  465</span>&#160;<span class="comment">  W - [in]</span></div>
<div class="line"><a name="l00466"></a><span class="lineno">  466</span>&#160;<span class="comment">    diagonal values in the singular value decomposition.</span></div>
<div class="line"><a name="l00467"></a><span class="lineno">  467</span>&#160;<span class="comment">  invW - [out]</span></div>
<div class="line"><a name="l00468"></a><span class="lineno">  468</span>&#160;<span class="comment">    The inverted diagonal is returned here.  invW may be the same</span></div>
<div class="line"><a name="l00469"></a><span class="lineno">  469</span>&#160;<span class="comment">    pointer as W.  If the input value of invW is not null, it must</span></div>
<div class="line"><a name="l00470"></a><span class="lineno">  470</span>&#160;<span class="comment">    point to an array of count doubles.  If the input value of</span></div>
<div class="line"><a name="l00471"></a><span class="lineno">  471</span>&#160;<span class="comment">    invW is null, heap storage will be allocated using onmalloc()</span></div>
<div class="line"><a name="l00472"></a><span class="lineno">  472</span>&#160;<span class="comment">    and the calling function must call onfree(invW).</span></div>
<div class="line"><a name="l00473"></a><span class="lineno">  473</span>&#160;<span class="comment">Remarks:</span></div>
<div class="line"><a name="l00474"></a><span class="lineno">  474</span>&#160;<span class="comment">  If the singular value decomposition were mathematically perfect, then</span></div>
<div class="line"><a name="l00475"></a><span class="lineno">  475</span>&#160;<span class="comment">  this function would be:</span></div>
<div class="line"><a name="l00476"></a><span class="lineno">  476</span>&#160;<span class="comment">    for (i = 0; i &lt; count; i++) </span></div>
<div class="line"><a name="l00477"></a><span class="lineno">  477</span>&#160;<span class="comment">      invW[i] = (W[i] != 0.0) ? 1.0/W[i] : 0.0;</span></div>
<div class="line"><a name="l00478"></a><span class="lineno">  478</span>&#160;<span class="comment">  Because the double precision arithmetic is not mathematically perfect,</span></div>
<div class="line"><a name="l00479"></a><span class="lineno">  479</span>&#160;<span class="comment">  very small values of W[i] may well be zero and this function makes</span></div>
<div class="line"><a name="l00480"></a><span class="lineno">  480</span>&#160;<span class="comment">  a reasonable guess as to when W[i] should be treated as zero.  </span></div>
<div class="line"><a name="l00481"></a><span class="lineno">  481</span>&#160;<span class="comment">Returns:</span></div>
<div class="line"><a name="l00482"></a><span class="lineno">  482</span>&#160;<span class="comment">  Number of non-zero elements in invW, which, in a mathematically perfect</span></div>
<div class="line"><a name="l00483"></a><span class="lineno">  483</span>&#160;<span class="comment">  situation, is the rank of Diag(W).</span></div>
<div class="line"><a name="l00484"></a><span class="lineno">  484</span>&#160;<span class="comment">*/</span></div>
<div class="line"><a name="l00485"></a><span class="lineno">  485</span>&#160;ON_DECL</div>
<div class="line"><a name="l00486"></a><span class="lineno">  486</span>&#160;<span class="keywordtype">int</span> ON_InvertSVDW(</div>
<div class="line"><a name="l00487"></a><span class="lineno">  487</span>&#160;  <span class="keywordtype">int</span> count, </div>
<div class="line"><a name="l00488"></a><span class="lineno">  488</span>&#160;  <span class="keyword">const</span> <span class="keywordtype">double</span>* W,</div>
<div class="line"><a name="l00489"></a><span class="lineno">  489</span>&#160;  <span class="keywordtype">double</span>*&amp; invW</div>
<div class="line"><a name="l00490"></a><span class="lineno">  490</span>&#160;  );</div>
<div class="line"><a name="l00491"></a><span class="lineno">  491</span>&#160; </div>
<div class="line"><a name="l00492"></a><span class="lineno">  492</span>&#160;<span class="comment">/*</span></div>
<div class="line"><a name="l00493"></a><span class="lineno">  493</span>&#160;<span class="comment">Description:</span></div>
<div class="line"><a name="l00494"></a><span class="lineno">  494</span>&#160;<span class="comment">  Solve a linear system of equations using the singular value decomposition.</span></div>
<div class="line"><a name="l00495"></a><span class="lineno">  495</span>&#160;<span class="comment">Parameters:</span></div>
<div class="line"><a name="l00496"></a><span class="lineno">  496</span>&#160;<span class="comment">  row_count - [in]</span></div>
<div class="line"><a name="l00497"></a><span class="lineno">  497</span>&#160;<span class="comment">    number of rows in matrix U</span></div>
<div class="line"><a name="l00498"></a><span class="lineno">  498</span>&#160;<span class="comment">  col_count - [in]</span></div>
<div class="line"><a name="l00499"></a><span class="lineno">  499</span>&#160;<span class="comment">    number of columns in matrix U</span></div>
<div class="line"><a name="l00500"></a><span class="lineno">  500</span>&#160;<span class="comment">  U - [in]</span></div>
<div class="line"><a name="l00501"></a><span class="lineno">  501</span>&#160;<span class="comment">    row_count X col_count matix.</span></div>
<div class="line"><a name="l00502"></a><span class="lineno">  502</span>&#160;<span class="comment">    See the remarks section for the definition of U.</span></div>
<div class="line"><a name="l00503"></a><span class="lineno">  503</span>&#160;<span class="comment">  invW - [in]</span></div>
<div class="line"><a name="l00504"></a><span class="lineno">  504</span>&#160;<span class="comment">    inverted DVD diagonal.</span></div>
<div class="line"><a name="l00505"></a><span class="lineno">  505</span>&#160;<span class="comment">    See the remarks section for the definition of invW.</span></div>
<div class="line"><a name="l00506"></a><span class="lineno">  506</span>&#160;<span class="comment">  V - [in]</span></div>
<div class="line"><a name="l00507"></a><span class="lineno">  507</span>&#160;<span class="comment">    col_count X col_count matrix.</span></div>
<div class="line"><a name="l00508"></a><span class="lineno">  508</span>&#160;<span class="comment">    See the remarks section for the definition of V.</span></div>
<div class="line"><a name="l00509"></a><span class="lineno">  509</span>&#160;<span class="comment">  B - [in]</span></div>
<div class="line"><a name="l00510"></a><span class="lineno">  510</span>&#160;<span class="comment">    An array of row_count values.</span></div>
<div class="line"><a name="l00511"></a><span class="lineno">  511</span>&#160;<span class="comment">  X - [out]</span></div>
<div class="line"><a name="l00512"></a><span class="lineno">  512</span>&#160;<span class="comment">    The solution array of col_count values is returned here.</span></div>
<div class="line"><a name="l00513"></a><span class="lineno">  513</span>&#160;<span class="comment">    If the input value of X is not null, it must point to an</span></div>
<div class="line"><a name="l00514"></a><span class="lineno">  514</span>&#160;<span class="comment">    array of col_count doubles.  If the input value of X is</span></div>
<div class="line"><a name="l00515"></a><span class="lineno">  515</span>&#160;<span class="comment">    null, heap storage will be allocated using onmalloc() and</span></div>
<div class="line"><a name="l00516"></a><span class="lineno">  516</span>&#160;<span class="comment">    the calling function must call onfree(X).</span></div>
<div class="line"><a name="l00517"></a><span class="lineno">  517</span>&#160;<span class="comment">Remarks:</span></div>
<div class="line"><a name="l00518"></a><span class="lineno">  518</span>&#160;<span class="comment">  If A*X = B is an m X n system of equations (m = row_count, n = col_count)</span></div>
<div class="line"><a name="l00519"></a><span class="lineno">  519</span>&#160;<span class="comment">  and A = U*Diag(W)*Transpose(V) is the singular value decompostion of A,</span></div>
<div class="line"><a name="l00520"></a><span class="lineno">  520</span>&#160;<span class="comment">  then a solution is X = V*Diag(1/W)*Transpose(U).</span></div>
<div class="line"><a name="l00521"></a><span class="lineno">  521</span>&#160;<span class="comment">Example:</span></div>
<div class="line"><a name="l00522"></a><span class="lineno">  522</span>&#160;<span class="comment"></span> </div>
<div class="line"><a name="l00523"></a><span class="lineno">  523</span>&#160;<span class="comment">          int m = row_count;</span></div>
<div class="line"><a name="l00524"></a><span class="lineno">  524</span>&#160;<span class="comment">          int n = col_count;</span></div>
<div class="line"><a name="l00525"></a><span class="lineno">  525</span>&#160;<span class="comment">          ON_Matrix A(m,n);</span></div>
<div class="line"><a name="l00526"></a><span class="lineno">  526</span>&#160;<span class="comment">          for (i = 0; i &lt; m; i++ ) for ( j = 0; j &lt; n; j++ )</span></div>
<div class="line"><a name="l00527"></a><span class="lineno">  527</span>&#160;<span class="comment">          {</span></div>
<div class="line"><a name="l00528"></a><span class="lineno">  528</span>&#160;<span class="comment">            A[i][j] = ...;</span></div>
<div class="line"><a name="l00529"></a><span class="lineno">  529</span>&#160;<span class="comment">          }</span></div>
<div class="line"><a name="l00530"></a><span class="lineno">  530</span>&#160;<span class="comment">          ON_SimpleArray&lt;double&gt; B(m);</span></div>
<div class="line"><a name="l00531"></a><span class="lineno">  531</span>&#160;<span class="comment">          for (i = 0; i &lt; m; i++ )</span></div>
<div class="line"><a name="l00532"></a><span class="lineno">  532</span>&#160;<span class="comment">          {</span></div>
<div class="line"><a name="l00533"></a><span class="lineno">  533</span>&#160;<span class="comment">            B[i] = ...;</span></div>
<div class="line"><a name="l00534"></a><span class="lineno">  534</span>&#160;<span class="comment">          }</span></div>
<div class="line"><a name="l00535"></a><span class="lineno">  535</span>&#160;<span class="comment"></span> </div>
<div class="line"><a name="l00536"></a><span class="lineno">  536</span>&#160;<span class="comment">          ON_SimpleArray&lt;double&gt; X; // solution returned here.</span></div>
<div class="line"><a name="l00537"></a><span class="lineno">  537</span>&#160;<span class="comment">          {</span></div>
<div class="line"><a name="l00538"></a><span class="lineno">  538</span>&#160;<span class="comment">            double** U = 0;</span></div>
<div class="line"><a name="l00539"></a><span class="lineno">  539</span>&#160;<span class="comment">            double* W = 0;</span></div>
<div class="line"><a name="l00540"></a><span class="lineno">  540</span>&#160;<span class="comment">            double** V = 0;</span></div>
<div class="line"><a name="l00541"></a><span class="lineno">  541</span>&#160;<span class="comment">            if ( ON_GetMatrixSVD(m,n,A.m,U,W,V) )</span></div>
<div class="line"><a name="l00542"></a><span class="lineno">  542</span>&#160;<span class="comment">            {</span></div>
<div class="line"><a name="l00543"></a><span class="lineno">  543</span>&#160;<span class="comment">              double* invW = 0;</span></div>
<div class="line"><a name="l00544"></a><span class="lineno">  544</span>&#160;<span class="comment">              int rankW = ON_InvertSVDW(n,W,W); // save invW into W</span></div>
<div class="line"><a name="l00545"></a><span class="lineno">  545</span>&#160;<span class="comment">              X.Reserve(n);</span></div>
<div class="line"><a name="l00546"></a><span class="lineno">  546</span>&#160;<span class="comment">              if ( ON_SolveSVD(m,n,U,W,V,B,X.Array()) )</span></div>
<div class="line"><a name="l00547"></a><span class="lineno">  547</span>&#160;<span class="comment">                X.SetCount(n);</span></div>
<div class="line"><a name="l00548"></a><span class="lineno">  548</span>&#160;<span class="comment">            }</span></div>
<div class="line"><a name="l00549"></a><span class="lineno">  549</span>&#160;<span class="comment">            onfree(U); // U allocated in ON_GetMatrixSVD()</span></div>
<div class="line"><a name="l00550"></a><span class="lineno">  550</span>&#160;<span class="comment">            onfree(W); // W allocated in ON_GetMatrixSVD()</span></div>
<div class="line"><a name="l00551"></a><span class="lineno">  551</span>&#160;<span class="comment">            onfree(V); // V allocated in ON_GetMatrixSVD()</span></div>
<div class="line"><a name="l00552"></a><span class="lineno">  552</span>&#160;<span class="comment">          }</span></div>
<div class="line"><a name="l00553"></a><span class="lineno">  553</span>&#160;<span class="comment"></span> </div>
<div class="line"><a name="l00554"></a><span class="lineno">  554</span>&#160;<span class="comment">          if ( n == X.Count() )</span></div>
<div class="line"><a name="l00555"></a><span class="lineno">  555</span>&#160;<span class="comment">          {</span></div>
<div class="line"><a name="l00556"></a><span class="lineno">  556</span>&#160;<span class="comment">            ... use solution</span></div>
<div class="line"><a name="l00557"></a><span class="lineno">  557</span>&#160;<span class="comment">          }  </span></div>
<div class="line"><a name="l00558"></a><span class="lineno">  558</span>&#160;<span class="comment">Returns:</span></div>
<div class="line"><a name="l00559"></a><span class="lineno">  559</span>&#160;<span class="comment">  True if input is valid and X[] was calculated. </span></div>
<div class="line"><a name="l00560"></a><span class="lineno">  560</span>&#160;<span class="comment">  False if input is not valid.</span></div>
<div class="line"><a name="l00561"></a><span class="lineno">  561</span>&#160;<span class="comment">*/</span></div>
<div class="line"><a name="l00562"></a><span class="lineno">  562</span>&#160;ON_DECL</div>
<div class="line"><a name="l00563"></a><span class="lineno">  563</span>&#160;<span class="keywordtype">bool</span> ON_SolveSVD(</div>
<div class="line"><a name="l00564"></a><span class="lineno">  564</span>&#160;  <span class="keywordtype">int</span> row_count,</div>
<div class="line"><a name="l00565"></a><span class="lineno">  565</span>&#160;  <span class="keywordtype">int</span> col_count,</div>
<div class="line"><a name="l00566"></a><span class="lineno">  566</span>&#160;  <span class="keywordtype">double</span> <span class="keyword">const</span> * <span class="keyword">const</span> * U,</div>
<div class="line"><a name="l00567"></a><span class="lineno">  567</span>&#160;  <span class="keyword">const</span> <span class="keywordtype">double</span>* invW,</div>
<div class="line"><a name="l00568"></a><span class="lineno">  568</span>&#160;  <span class="keywordtype">double</span> <span class="keyword">const</span> * <span class="keyword">const</span> * V,</div>
<div class="line"><a name="l00569"></a><span class="lineno">  569</span>&#160;  <span class="keyword">const</span> <span class="keywordtype">double</span>* B,</div>
<div class="line"><a name="l00570"></a><span class="lineno">  570</span>&#160;  <span class="keywordtype">double</span>*&amp; X</div>
<div class="line"><a name="l00571"></a><span class="lineno">  571</span>&#160;  );</div>
<div class="line"><a name="l00572"></a><span class="lineno">  572</span>&#160;  </div>
<div class="line"><a name="l00573"></a><span class="lineno">  573</span>&#160; </div>
<div class="line"><a name="l00574"></a><span class="lineno">  574</span>&#160;<span class="comment">/*</span></div>
<div class="line"><a name="l00575"></a><span class="lineno">  575</span>&#160;<span class="comment">Description:</span></div>
<div class="line"><a name="l00576"></a><span class="lineno">  576</span>&#160;<span class="comment">  Perform simple row reduction on a matrix.  If A is square, positive</span></div>
<div class="line"><a name="l00577"></a><span class="lineno">  577</span>&#160;<span class="comment">  definite, and really really nice, then the returned B is the inverse</span></div>
<div class="line"><a name="l00578"></a><span class="lineno">  578</span>&#160;<span class="comment">  of A.  If A is not positive definite and really really nice, then it</span></div>
<div class="line"><a name="l00579"></a><span class="lineno">  579</span>&#160;<span class="comment">  is probably a waste of time to call this function.</span></div>
<div class="line"><a name="l00580"></a><span class="lineno">  580</span>&#160;<span class="comment">Parameters:</span></div>
<div class="line"><a name="l00581"></a><span class="lineno">  581</span>&#160;<span class="comment">  row_count - [in]</span></div>
<div class="line"><a name="l00582"></a><span class="lineno">  582</span>&#160;<span class="comment">  col_count - [in]</span></div>
<div class="line"><a name="l00583"></a><span class="lineno">  583</span>&#160;<span class="comment">  zero_pivot - [in]</span></div>
<div class="line"><a name="l00584"></a><span class="lineno">  584</span>&#160;<span class="comment">    absolute values &lt;= zero_pivot are considered to be zero</span></div>
<div class="line"><a name="l00585"></a><span class="lineno">  585</span>&#160;<span class="comment">  A - [in/out]</span></div>
<div class="line"><a name="l00586"></a><span class="lineno">  586</span>&#160;<span class="comment">    A row_count X col_count matrix.  Input is the matrix to be</span></div>
<div class="line"><a name="l00587"></a><span class="lineno">  587</span>&#160;<span class="comment">    row reduced.  The calculation destroys A, so output A is garbage.</span></div>
<div class="line"><a name="l00588"></a><span class="lineno">  588</span>&#160;<span class="comment">  B - [out]</span></div>
<div class="line"><a name="l00589"></a><span class="lineno">  589</span>&#160;<span class="comment">    A a row_count X row_count matrix. That records the row reduction.</span></div>
<div class="line"><a name="l00590"></a><span class="lineno">  590</span>&#160;<span class="comment">  pivots - [out]</span></div>
<div class="line"><a name="l00591"></a><span class="lineno">  591</span>&#160;<span class="comment">    minimum and maximum absolute values of pivots.</span></div>
<div class="line"><a name="l00592"></a><span class="lineno">  592</span>&#160;<span class="comment">Returns:</span></div>
<div class="line"><a name="l00593"></a><span class="lineno">  593</span>&#160;<span class="comment">  Rank of A.  If the returned value &lt; min(row_count,col_count),</span></div>
<div class="line"><a name="l00594"></a><span class="lineno">  594</span>&#160;<span class="comment">  then a zero pivot was encountered.</span></div>
<div class="line"><a name="l00595"></a><span class="lineno">  595</span>&#160;<span class="comment">  If C = input value of A, then B*C = (I,*)</span></div>
<div class="line"><a name="l00596"></a><span class="lineno">  596</span>&#160;<span class="comment">*/</span></div>
<div class="line"><a name="l00597"></a><span class="lineno">  597</span>&#160;ON_DECL</div>
<div class="line"><a name="l00598"></a><span class="lineno">  598</span>&#160;<span class="keywordtype">int</span> ON_RowReduce( </div>
<div class="line"><a name="l00599"></a><span class="lineno">  599</span>&#160;          <span class="keywordtype">int</span> row_count, </div>
<div class="line"><a name="l00600"></a><span class="lineno">  600</span>&#160;          <span class="keywordtype">int</span> col_count,</div>
<div class="line"><a name="l00601"></a><span class="lineno">  601</span>&#160;          <span class="keywordtype">double</span> zero_pivot,</div>
<div class="line"><a name="l00602"></a><span class="lineno">  602</span>&#160;          <span class="keywordtype">double</span>** A, </div>
<div class="line"><a name="l00603"></a><span class="lineno">  603</span>&#160;          <span class="keywordtype">double</span>** B, </div>
<div class="line"><a name="l00604"></a><span class="lineno">  604</span>&#160;          <span class="keywordtype">double</span> pivots[2] </div>
<div class="line"><a name="l00605"></a><span class="lineno">  605</span>&#160;          );</div>
<div class="line"><a name="l00606"></a><span class="lineno">  606</span>&#160; </div>
<div class="line"><a name="l00607"></a><span class="lineno">  607</span>&#160;<span class="preprocessor">#endif</span></div>
<div class="ttc" id="aclass_o_n__3d_point_html"><div class="ttname"><a href="class_o_n__3d_point.html">ON_3dPoint</a></div><div class="ttdef"><b>Definition:</b> opennurbs_point.h:419</div></div>
<div class="ttc" id="aclass_o_n___matrix_html"><div class="ttname"><a href="class_o_n___matrix.html">ON_Matrix</a></div><div class="ttdef"><b>Definition:</b> opennurbs_matrix.h:23</div></div>
<div class="ttc" id="aclass_o_n___simple_array_html"><div class="ttname"><a href="class_o_n___simple_array.html">ON_SimpleArray&lt; double &gt;</a></div></div>
<div class="ttc" id="aclass_o_n___xform_html"><div class="ttname"><a href="class_o_n___xform.html">ON_Xform</a></div><div class="ttdef"><b>Definition:</b> opennurbs_xform.h:29</div></div>
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